Factorials outrun exponentials, exponentials outrun powers, and powers outrun logarithms.Suggest a slogan
For real sequences \(a(n)\), \(b(n)\) that are positive for all sufficiently large \(n\), write \(a\prec b\) if \(a(n)/b(n)\to 0\) as \(n\to\infty\). For every real \(c>1\), \[ 1 \prec \log(\log n) \prec \log n \prec n \prec n\log n \prec n^{c} \prec c^n \prec n!. \]
How do $n!$, $c^n$, $n^c$, $n\log n$, $n$ and $\log n$ rank by growth rate?